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In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in ) bounded by C. It is the two-dimensional special case of Stokes' theorem (surface in ). In one dimension, it is equivalent to the fundamental theorem of calculus.
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
Download as PDF; Printable version; In other projects ... In mathematics, Green formula may refer to: Green's theorem in integral calculus; Green's identities in ...
is the derivative of the Green's function along the inward-pointing unit normal vector ^. The integration is performed on the boundary, with measure d s {\displaystyle ds} . The function ν ( s ) {\displaystyle \nu (s)} is given by the unique solution to the Fredholm integral equation of the second kind,
Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George Green. Green's treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point: to this function Green gave the name ...
In mathematics, the definite integral ()is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.
In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.